When I first read the Harry Potter books at age 8 Hermione's name always vexed me. Now after seeing the movies I read her name as spoken in the movies, but did Rowling ever give us a phonetic pronunciation for her name?
JKR pronounces it exactly as it's pronounced in the films
and from the FAQ on JKR's old website (along with a handy guide for how to remember it)
Q. How do you pronounce 'Hermione?'
A. Her (as in 'her brain is bigger than everyone else's') + my (as in 'my brain isn't as big as that') + oh (as in 'oh, for a brain that size') + knee (as in 'I've bruised my knee').
and from this interview
Q. This is probably a very American question, but how do you pronounce "Hermione"?
JKR. It's pronounced: Her-my-oh-nee.
And from Harry Potter and the Goblet of Fire
Hermione was now teaching Krum to say her name properly; he kept calling her ‘Hermy-own’.
‘Her – my – oh – nee,’ she said, slowly and clearly.
‘Herm – own – ninny.’
‘Close enough,’ she said, catching Harry’s eye and grinning.
A phonetic pronunciation of her name is given in Harry Potter and the Goblet of Fire. Ron says her name while his mouth is full of food, and it is rendered as four syllables.
‘Oh, c’mon, ’Er-my-knee,’ said Ron, accidentally spraying Harry with bits of Yorkshire pudding. ‘Oops – sorry, ’Arry –’ He swallowed. ‘You won’t get them sick leave by starving yourself!’
The presence of this passage in the book, along with the earlier one where it is pointed out that the reading pronunciation hermy-own is wrong, are presumably not merely incidental. Rather, the author and the publisher probably realized that lots of fans were not interpreting Hermione's name the way the author intended.
Hermione is a real name, taken from ancient Greek and along with other classical names was fairly common in English speaking countries around the turn of the 19th century, much like Penelope (still common) and Persephone.
Greek derived names are usually rendered into English pronouncing every letter so the conventional pronunciation would be Her-my-uh-knee with the main stress on the second syllable here the third syllable could easily vary between 'uh' and 'er' depending on the speaker without much noticeable difference as in miner, someone with very precise diction might render it as 'oh'.
Although with some English regional accents this would be more like Uh-my-knee or Er-my-knee
It's Her-my-oh-knee, as proved in GOF (Goblet of Fire) at the Yule Ball. Hermione was teaching Viktor Krum how to pronounce her name (he keeps calling her Hermy-own). Based on the spelling, it could be pronounced Her-my-oh-knee, Her-me-one, Her-me-own, Her-my-one, Her-mi-on, Her-me-on, and Her-me-own, but not Her-my-knee (I know, that's a lot.)